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Critical Damping

Ask
For the control system in the given figure, the value of K for critical damping is
(a) 1
(b) 5.125
(c) 6.831
(d) 10



Answer is ( b )
Concept
We should first consider simplifying the block - diagram in the question as:



Combining the two cascaded blocks , we get : G(s) = (K)(\frac{2}{s^2+7s+2})

A closed loop control system having a form as below:



Thus , the above diagram simplifies to \frac{C(s)}{R(s)} = \frac{(\frac{2K}{s^2+7s+2})}{1+(\frac{2K}{s^2+7s+2})}

\implies \frac{C(s)}{R(s)}=  \frac{2K}{s^2+7s+2+2K}

Now to Solve the Above Equation we need to understand the characteristic equation of a second - order system.

In general a second - order control system equation is given by :
\frac{C(s)}{R(s)}= \frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}

where,
\zeta = damping ratio ; \omega_n = natural frequency
The characteristic Equation of the system is given by :s^2+2\zeta\omega_n s+\omega_n^2 = 0
Comparing this equation to an algebraic equation of the form ax^2+bx+c=0
The roots of the above equation are x= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
For a critical damping case b^2 - 4ac= 0 , \implies x= \frac{-b}{2a}

The characteristic equation for the given problem : s^2+7s+2+2K = 0
As for critical damping case b^2 - 4ac= 0
\implies  7^2 - 4(1)(2+2K)= 0
\implies  K=41/8 = 5.125











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